GARCH Toolbox

Equality Constraints and Initial Parameter Estimates

This section highlights some important points regarding equality constraints and initial parameter estimates in the GARCH Toolbox. It discusses:

• Complete Model Specification
• Empty Fix Fields
• Number of Equality Constraints

Complete Model Specification

To set equality constraints during estimation, you must provide a complete model specification. The only flexibility in this regard is that you can decouple the model specification for the conditional mean from the model specification for the conditional variance.

The following example demonstrates an attempt to set equality constraints for an incomplete conditional mean model and a complete variance model. Create an ARMA(1,1)/GARCH(1,1) specification structure for conditional mean and variance models, respectively.

• spec = garchset('R', 1, 'M', 1, 'P', 1, 'Q', 1);
  spec = garchset(spec, 'C', 0, 'AR', 0.5, 'FixAR', 1);
  spec = garchset(spec, 'K', 0.0005, 'GARCH', 0.8, 'ARCH', 0.1, 
  'FixGARCH', 1)
  spec = 
           Comment: 'Mean: ARMAX(1,1,?); Variance: GARCH(1,1)'
                 R: 1
                 M: 1
                 P: 1
                 Q: 1
      Distribution: 'Gaussian'
                 C: 0
                AR: 0.5000
                MA: []
           Regress: []
                 K: 5.0000e-004
             GARCH: 0.8000
              ARCH: 0.1000
              FixC: []
             FixAR: 1
             FixMA: []
        FixRegress: []
              FixK: []
          FixGARCH: 1
           FixARCH: []
      Optimization: [1x1 struct]
  

The conditional mean model is incomplete because the MA field is still empty. Since the requested ARMA(1,1) model is an incomplete conditional mean specification, garchfit ignores the C, AR, and FixAR fields, computes initial parameter estimates, and overwrites any existing parameters in the incomplete conditional mean specification. It also estimates all conditional mean parameters (i.e., C, AR, and MA) and ignores the request to constrain the AR parameter.

However, since the structure explicitly sets all fields in the conditional variance model, garchfit uses the specified values of K and ARCH as initial estimates subject to further refinement, and holds the GARCH parameter at 0.8 throughout the optimization process.

Empty Fix Fields

Any fix field that you leave empty ([]), is equivalent to a vector of zeros of compatible length. That is, when garchfit encounters an empty fix field, it automatically estimates the corresponding parameter. For example, the following specification structures produce the same GARCH(1,1) estimation results.

• spec1 = garchset('K', 0.005, 'GARCH', 0.8, 'ARCH', 0.1, 
  'FixGARCH', 0, 'FixARCH', 0)
  spec2 = garchset('K', 0.005, 'GARCH', 0.8, 'ARCH', 0.1)
  

Number of Equality Constraints

Avoid setting several equality constraints simultaneously. Although the ability to set equality constraints is both convenient and useful, equality constraints complicate the estimation process. For example, if you really want to estimate a GARCH(1,1) model, then specify a GARCH(1,1) model instead of a more elaborate model with numerous constraints.

Equality Constraints and Parameter Significance

Recommendations and Suggestions